Although fractal modeling of rough surfaces in contact was motivated by inadequacies in the early theory and applications of tribology, there is a plethora of problems that can be benefited from this type of modeling. Usually these are problems that require an estimate of material properties across interfaces between dissimilar materials, although many other areas such as sonic and electromagnetic scattering as well as image processing applications can be benefited by fractal modeling. Within the context of material science, it has already been demonstrated that such a modeling approach is useful for contact of deformable bodies, temperature distribution, friction, thermal contact conductance, and electric resistance.
The Weierstrass-Mandelbrot (W-M) function is employed for surface parametrization and in one approach is a multivariate analytic generalization of its univariate version and has wide applicability in all applications requiring an analytical description of a rough surface. There has been a considerable effort in establishing methodologies of determining just a single parameter describing surface representations and that is the fractal dimension.
The most widely used method for the determination of the fractal dimension—as one of the parameters characterizing the surface under investigation—, regardless of the particular analytical model, is the power spectrum method. However, beyond the obvious weakness that only one of the parameters of the fractal surface can be identified by this method (i.e. the fractal dimension), the method has proven not to be accurate enough and it also enforces assumptions requiring a priori definition of the rest of the parameters that can be very restrictive or not always true.
Surface Modeling
It has been established that a W-M function is s very rich analytical representation that can model surface topographies of fractal nature such as material surfaces at small scales. This seems to be true especially because of the properties of continuity, non-differentiability and self affinity of specific types of fractals, that are also desired properties of surface topographies. The surface power spectra obeys a power-law relationship over a wide range of frequencies, because the surface topographies resemble a random process. Such a surface can be represented by a complex function W as:
                              W          ⁡                      (            x            )                          =                              ∑                          n              =                              -                ∞                                      ∞                    ⁢                                                    γ                                                      (                                          D                      -                      2                                        )                                    ⁢                  n                                            ⁡                              (                                  1                  -                                      ⅇ                                          ⅈ                      ⁢                                                                                          ⁢                                              γ                                                  n                          x                                                                                                                    )                                      ⁢                          ⅇ                              ⅈ                ⁢                                                                  ⁢                                  ϕ                  n                                                                                        (        1        )            where x is a real variable. A two dimensional profile can be obtained from the real part of Eq. 1:
                              z          ⁡                      (            x            )                          =                              Re            ⁡                          [                              W                ⁡                                  (                  x                  )                                            ]                                =                                    ∑                              n                =                                  -                  ∞                                            ∞                        ⁢                                                            γ                                                            (                                              D                        -                        2                                            )                                        ⁢                    n                                                  ⁡                                  [                                                            cos                      ⁢                                                                                          ⁢                                              ϕ                        n                                                              -                                          cos                      ⁡                                              (                                                                                                            γ                              n                                                        ⁢                            x                                                    +                                                      ϕ                            n                                                                          )                                                                              ]                                            .                                                          (        2        )            
The relevant parameters here are defined as follows: D is the fractal dimension (1<D<2 for line profiles), φn is a random phase that is used to prevent coincidence of different phases, n is the frequency index and γ is a parameter that controls the density of the frequencies and must be greater than 1; γ usually takes values in the vicinity of 1.5 because of surface flatness and frequency distribution density considerations. The later though has been recently debated and only the requirement γ>1 was considered as a valid assumption.
A three-dimensional fractal surface that exhibits randomness is the two-variable W-M function that is given by:
                              z          ⁡                      (                          r              ,              θ                        )                          =                                            (                                                ln                  ⁢                                                                          ⁢                  γ                                M                            )                                ⁢                                    ∑                              m                =                1                            M                        ⁢                                          ∑                                  n                  =                                      -                    ∞                                                  ∞                            ⁢                                                                    (                                          k                      ⁢                                                                                          ⁢                                              γ                        n                                                              )                                                        D                    -                    3                                                  ⁢                                  {                                                            cos                      ⁢                                                                                          ⁢                                              ϕ                        mn                                                              -                                          cos                      ⁡                                              [                                                                              k                            ⁢                                                                                                                  ⁢                                                          γ                              n                                                        ⁢                            r                            ⁢                                                                                                                  ⁢                                                          cos                              ⁡                                                              (                                                                  θ                                  -                                                                      α                                    m                                                                                                  )                                                                                                              +                                                      ϕ                            mn                                                                          ]                                                                              }                                                                                        (        3        )            
where M is the number of superimposed ridges, D now takes values between 2 and 3, αm is an arbitrary angle that is used to offset the ridges in the azimuthal direction and is equal to πm/M for equally offset ridges. k is the wave number and is given by: k=2 π/L, L is the size of the sample. In practice the upper limit of n is not infinite and is given by:N=nmax=int[log(L/Lc)/log γ]  (4)
with Lc being a cut-off wavelength, typically defined either by the highest sampling frequency, or by a physical barrier like the interatomic distance of the surface atoms. Since the lowest frequency is 1/L, the lowest limit of n can be set equal to 0. Finally the cartesian coordinates (x,y) are mapped to polar coordinates (r,θ) according to:
                              r          =                                                    x                2                            +                              y                2                                                    ,                  θ          =                                    tan                              -                1                                      ⁡                          (                              y                x                            )                                                          (        5        )            
If we substitute the previous relationships of Am, αm, r, θ, k and the limits of n, in Eq. 3 we get:
                              z          ⁡                      (                          x              ,              y                        )                          =                                            A              ⁡                              (                                  L                                      2                    ⁢                    π                                                  )                                                    3              -              D                                ⁢                                                    ln                ⁢                                                                  ⁢                γ                            M                                ⁢                                    ∑                              m                =                1                            M                        ⁢                                          ∑                                  n                  =                  0                                N                            ⁢                                                          ⁢                                                γ                                                            (                                              D                        -                        3                                            )                                        ⁢                    n                                                  ⁢                                  {                                      cos                    ⁢                                                                                  ⁢                                                                  ϕ                        mn                                            ·                                              --                                                  cos                          ⁡                                                      [                                                                                                                                                                2                                    ⁢                                    π                                    ⁢                                                                                                                                                  ⁢                                                                          γ                                      n                                                                        ⁢                                                                                                                                                            x                                          2                                                                                +                                                                                  y                                          2                                                                                                                                                                                      L                                                                ⁢                                                                  cos                                  ⁡                                                                      (                                                                                                                                                            tan                                                                                      -                                            1                                                                                                                          ⁡                                                                                  (                                                                                      y                                            x                                                                                    )                                                                                                                    -                                                                                                                        π                                          ⁢                                                                                                                                                                          ⁢                                          m                                                                                M                                                                                                              )                                                                                                                              +                                                              ϕ                                mn                                                                                      ]                                                                                                                                                                                                      (        6        )            Parameter A can be substituted by 2π(2 π/G)2-D, and therefore Eq. 6 becomes:
                              z          ⁡                      (                          x              ,              y                        )                          =                                            L              ⁡                              (                                  G                  L                                )                                                    D              -              2                                ⁢                                                    ln                ⁢                                                                  ⁢                γ                            M                                ⁢                                    ∑                              m                =                1                            M                        ⁢                                          ∑                                  n                  =                  0                                N                            ⁢                                                          ⁢                                                γ                                                            (                                              D                        -                        3                                            )                                        ⁢                    n                                                  ⁢                                  {                                      cos                    ⁢                                                                                  ⁢                                                                  ϕ                        mn                                            ·                                              --                                                  cos                          ⁡                                                      [                                                                                                                                                                2                                    ⁢                                    π                                    ⁢                                                                                                                                                  ⁢                                                                          γ                                      n                                                                        ⁢                                                                                                                                                            x                                          2                                                                                +                                                                                  y                                          2                                                                                                                                                                                      L                                                                ⁢                                                                  cos                                  ⁡                                                                      (                                                                                                                                                            tan                                                                                      -                                            1                                                                                                                          ⁡                                                                                  (                                                                                      y                                            x                                                                                    )                                                                                                                    -                                                                                                                        π                                          ⁢                                                                                                                                                                          ⁢                                          m                                                                                M                                                                                                              )                                                                                                                              +                                                              ϕ                                mn                                                                                      ]                                                                                                                                                                                                      (        7        )            
The parameter G is independent of the frequency and is referred to as the fractal roughness.
Although the surface representation of Eq. 7 is in a generally convenient form for computations and phenomenological observations, it is still not in a form that can be used to identify the phases φmn. In order to achieve phase identification for a given set of topographic or elevation data, we need to use an expression that decouples the phases from the other variables in the function. Such a refactored representation exists and in the 2 dimensional case is expressed by the complex function:
                                                                        W                ⁡                                  (                  r                  )                                            =                            ⁢                              W                ⁡                                  (                                      r                    ,                    θ                                    )                                                                                                        =                            ⁢                                                                                          ln                      ⁢                                                                                          ⁢                      γ                                        M                                                  ⁢                                                      ∑                                          m                      =                      1                                        M                                    ⁢                                                            A                      m                                        ⁢                                                                  ∑                                                  n                          =                                                      -                            ∞                                                                          ∞                                            ⁢                                                                        (                                                      –ⅇ                                                          ⅈ                              ⁢                                                                                                                          ⁢                                                              k                                0                                                            ⁢                                                              γ                                n                                                            ⁢                              r                              ⁢                                                                                                                          ⁢                                                              cos                                ⁡                                                                  (                                                                      θ                                    -                                                                          a                                      m                                                                                                        )                                                                                                                                              )                                                ⁢                                                                                                            ⅇ                                                              ⅈϕ                                mn                                                                                      ⁡                                                          (                                                                                                k                                  0                                                                ⁢                                                                  γ                                  n                                                                                            )                                                                                                            D                            -                            3                                                                                                                                                                                                      (        8        )            
Performing the same substitutions with those in Eqs. 3, 6 and 7, the previous relationship takes the refactored form:
                              W          ⁡                      (                          r              ,              θ                        )                          =                                            L              ⁡                              (                                  G                  L                                )                                                    D              -              2                                ⁢                                                    ln                ⁢                                                                  ⁢                γ                            M                                ⁢                                    ∑                              m                =                1                            M                        ⁢                                          ∑                                  n                  =                  0                                N                            ⁢                                                                    γ                                                                  (                                                  D                          -                          3                                                )                                            ⁢                      n                                                        ⁡                                      (                                          1                      ⁢                                              –ⅇ                                                  ⅈ                          ⁢                                                                                                          ⁢                                                                                    2                              ⁢                              π                                                        L                                                    ⁢                                                      γ                            n                                                    ⁢                          r                          ⁢                                                                                                          ⁢                                                      cos                            ⁡                                                          (                                                              θ                                -                                                                  a                                  m                                                                                            )                                                                                                                                            )                                                  ⁢                                  ⅇ                                      ⅈϕ                    mn                                                                                                          (        9        )            
The parameters in this equation have the exact same meaning as the parameters in Eq. 3.
Next, assume that for a given set of parameters, a surface is described by Eq. 9. For another set of the D and G parameters D′ and G′ we seek to calculate the new phases, so that the new and the original surfaces coincide:W(r,θ)=W′(r,θ),∀r,θεR  (10)or:
                                              ⁢                                                            L                ⁡                                  (                                      G                    L                                    )                                                            D                -                2                                      ⁢                                                            ln                  ⁢                                                                          ⁢                  γ                                M                                      ⁢                                          ∑                                  m                  =                  1                                M                            ⁢                                                          ⁢                                                ∑                                      n                    =                    0                                    N                                ⁢                                                                            γ                                                                        (                                                      D                            -                            3                                                    )                                                ⁢                        n                                                              (                                          1                      -                                              ⅇ                                                  ⅈ                          ⁢                                                                                    2                              ⁢                              π                                                        L                                                    ⁢                                                      γ                            n                                                    ⁢                                                      rcos                            ⁡                                                          (                                                              θ                                -                                                                  a                                  m                                                                                            )                                                                                                                                            )                                    ⁢                                      ⅇ                                          ⅈϕ                      mn                                                                                                    ==                                                    L                ⁡                                  (                                                            G                      ′                                        L                                    )                                                                              D                  ′                                -                2                                      ⁢                                                            ln                  ⁢                                                                          ⁢                  γ                                M                                      ⁢                                          ∑                                  m                  =                  1                                M                            ⁢                                                          ⁢                                                          ⁢                                                ∑                                      n                    =                    0                                    N                                ⁢                                                                            γ                                                                        (                                                                                    D                              ′                                                        -                            3                                                    )                                                ⁢                        n                                                              (                                          1                      -                                              ⅇ                                                  ⅈ                          ⁢                                                                                    2                              ⁢                              π                                                        L                                                    ⁢                                                      γ                            n                                                    ⁢                                                      rcos                            ⁡                                                          (                                                              θ                                -                                                                  a                                  m                                                                                            )                                                                                                                                            )                                    ⁢                                      ⅇ                                          ⅈϕ                      mn                      ′                                                                                                                              (        11        )            For Eq. 11 to hold for all r, θεR it must also hold that all the added in parameters of the sums be equal. Since the
      c    ⁡          (              r        ,        θ            )        =      (          1      -              ⅇ                  ⅈ          ⁢                                    2              ⁢              π                        L                    ⁢                      γ            n                    ⁢                      rcos            ⁡                          (                              θ                -                                  a                  m                                            )                                            )  expressions on the left and right side of Eq. 11 are equal, it must therefore hold:
                                                        L              ⁡                              (                                  G                  L                                )                                                    D              -              2                                ⁢                                                    ln                ⁢                                                                  ⁢                γ                            M                                ⁢                      γ                                          (                                  D                  -                  3                                )                            ⁢              n                                ⁢                      ⅇ                          ⅈϕ              mn                                      =                                            L              ⁡                              (                                                      G                    ′                                    L                                )                                                                    D                ′                            -              2                                ⁢                                                    ln                ⁢                                                                  ⁢                γ                            M                                ⁢                      γ                                          (                                                      D                    ′                                    -                  3                                )                            ⁢              n                                ⁢                      ⅇ                          ⅈϕ              mn              ′                                                          (        12        )            Solving Eq. 12 for φ′mn:
                              ϕ          mn          ′                =                              2            ⁢            π            ⁢                                                  ⁢            v                    -                      i            ⁢                                                  ⁢                          ln              [                                                                                          ⅇ                                              ⅈϕ                        mn                                                              ⁡                                          (                                              G                        L                                            )                                                                            D                    -                    2                                                  ⁢                                                                            γ                                              n                        ⁡                                                  (                                                      D                            -                                                          D                              ′                                                                                )                                                                                      ⁡                                          (                                                                        G                          ′                                                L                                            )                                                                            2                    -                                          D                      ′                                                                                  ]                                                          (        13        )            
with vεZ an arbitrary number. Clearly, the fact that we can always find a new set of values for the phases, for any combination of the fractal parameters D and G indicates that we can always find a surface, independent of the magnitudes of those two parameters. From a characterization perspective this means that these two parameters can be selected to be known, since the phases can adjust for different choices of D and G. This finding has serious implications on all physics-based models that are based on parameters D and G, because we have demonstrated that alternative values for D and G can be used if an alternative but consistent set of phases is established.
It would therefore be desirable to provide a method for determining all of the critical parameters involved in the full specification of the W-M function.